Evaluate: `sqrt(5+12i)`

Answer

To evaluate \( \sqrt{5 + 12i} \), we will follow a systematic approach using the properties of complex numbers.

### Step 1: Let \( z = \sqrt{5 + 12i} \)

We can express \( z \) in terms of its real and imaginary parts:
\[ z = x + yi \]
where \( x \) is the real part and \( y \) is the imaginary part.

### Step 2: Square both sides

Squaring both sides gives us:
\[ z^2 = 5 + 12i \]
Substituting \( z = x + yi \) into the equation, we have:
\[ (x + yi)^2 = 5 + 12i \]
Expanding the left side:
\[ x^2 + 2xyi - y^2 = 5 + 12i \]

### Step 3: Separate real and imaginary parts

From the equation \( x^2 - y^2 + 2xyi = 5 + 12i \), we can equate the real and imaginary parts:
1. Real part: \( x^2 - y^2 = 5 \)
2. Imaginary part: \( 2xy = 12 \)

### Step 4: Solve the equations

From the imaginary part equation \( 2xy = 12 \), we can simplify it to:
\[ xy = 6 \quad \text{(1)} \]

Now we have a system of equations:
1. \( x^2 - y^2 = 5 \quad \text{(2)} \)
2. \( xy = 6 \quad \text{(1)} \)

### Step 5: Express \( y \) in terms of \( x \)

From equation (1), we can express \( y \) as:
\[ y = \frac{6}{x} \]

### Step 6: Substitute \( y \) into equation (2)

Substituting \( y \) into equation (2):
\[ x^2 - \left(\frac{6}{x}\right)^2 = 5 \]
This simplifies to:
\[ x^2 - \frac{36}{x^2} = 5 \]
Multiplying through by \( x^2 \) to eliminate the fraction:
\[ x^4 - 5x^2 - 36 = 0 \]

### Step 7: Let \( u = x^2 \)

Let \( u = x^2 \), then we have a quadratic equation:
\[ u^2 - 5u - 36 = 0 \]

### Step 8: Solve the quadratic equation

Using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ u = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ u = \frac{5 \pm \sqrt{25 + 144}}{2} \]
\[ u = \frac{5 \pm \sqrt{169}}{2} \]
\[ u = \frac{5 \pm 13}{2} \]

Calculating the two possible values:
1. \( u = \frac{18}{2} = 9 \) (so \( x^2 = 9 \) and \( x = 3 \))
2. \( u = \frac{-8}{2} = -4 \) (not valid since \( u = x^2 \) cannot be negative)

### Step 9: Find \( y \)

Now substituting \( x = 3 \) back into equation (1):
\[ y = \frac{6}{3} = 2 \]

### Step 10: Write the final result

Thus, we have:
\[ z = 3 + 2i \]
Therefore, the value of \( \sqrt{5 + 12i} \) is:
\[ \sqrt{5 + 12i} = 3 + 2i \]